On the distribution of distance signless Laplacian eigenvalues with given independence and chromatic number

For a connected graph G of order n, let D(G) be the distance matrix and T r(G) be the diagonal matrix of vertex transmissions of G.The distance signless Laplacian (dsL, for short) matrix of G is defined as, and the corresponding eigenvalues are the dsL eigenvalues of G.For an interval I, let m D Q (G) I denote the number of dsL eigenvalues of G lying in the interval I.In this paper, for some prescribed interval I, we obtain bounds for m D Q (G) I in terms of the independence number α and the chromatic number χ of G. Furthermore, we provide lower bounds of ∂ Q 1 (G), the dsL spectral radius, for certain families of graphs in terms of the order n and the independence number α, or the chromatic number χ.